OFFSET
1,1
COMMENTS
Conjecture: a(n) exists for any n > 0. In general, for any nonzero integer m and positive integer n, the set {prime(k*n)+m: k = 1,2,3,...} always contains three distinct elements x, y and z with x*y = z.
REFERENCES
Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(1) = 6 since prime(6*1)-1 = 12 = 2*6 = (prime (2*1)-1)*(prime(4*1)-1).
a(4) = 71 since prime(71*4)-1 = 1860 = 6*310 = (prime(1*4)-1)*(prime(16*4)-1).
a(41) = 36235 since prime(36235*41)-1 = 23634312 = 676*34962 = (prime(3*41)-1)*(prime(91*41)-1).
a(69) = 64999 since prime(64999*69)-1 = 76643820 = 4590*16698 = (prime(9*69)-1)*(prime(28*69)-1).
a(77) = 137789 since prime(137789*77)-1 = 191037600 = 2028*94200 = (prime(4*77)-1)*(prime(118*77)-1).
a(99) = 167708 since prime(167708*99)-1 = 306849088 = 10528*29146 = (prime(13*99)-1)*(prime(32*99)-1).
a(189) = 951492 since prime(951492*189)-1 = 3776304996 = 4126*915246 = (prime(3*189)-1)*(prime(383*189)-1).
MATHEMATICA
Dv[n_]:=Divisors[Prime[n]-1]
L[n_]:=Length[Dv[n]]
P[k_, n_, i_]:=PrimeQ[Part[Dv[k*n], i]+1]&&Mod[PrimePi[Part[Dv[k*n], i]+1], n]==0
Do[k=0; Label[bb]; k=k+1; Do[If[P[k, n, i]&&P[k, n, L[k*n]-i+1], Goto[aa]], {i, 2, L[k*n]/2}]; Goto[bb]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 50}]
PROG
(PARI) a(n)={my(i, j, k=3); while(1, for(j=2, k-1, for(i=1, j-1, if(prime(k*n) - 1 == (prime(i*n)-1)*(prime(j*n)-1), break(3)); )); k++); return(k); }
main(size)={return(vector(size, n, a(n))); } /* Anders Hellström, Jul 13 2015 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 13 2015
STATUS
approved